10-46
10-71 A spherical container filled with iced water is subjected to convection and radiation heat transfer at
its outer surface. The rate of heat transfer and the amount of ice that melts per day are to be determined.
Assumptions 1 Heat transfer is steady since the specified thermal conditions at the boundaries do not
change with time. 2 Heat transfer is one-dimensional since there is thermal symmetry about the midpoint. 3
Thermal conductivity is constant.
Properties The thermal conductivity of steel is given to be k = 15 W/m˜°C. The heat of fusion of water at 1
atm is hif 333.7 kJ/kg . The outer surface of the tank is black and thus its emissivity is H = 1.
Analysis (a) The inner and the outer surface areas of sphere are
Ai SDi 2 S (8 m) 2 201.06 m 2
Ao SDo 2 S (8.03 m) 2 202.57 m 2
We assume the outer surface temperature T2 to be 5°C after comparing convection heat transfer coefficients
at the inner and the outer surfaces of the tank. With this assumption, the radiation heat transfer coefficient
can be determined from
hrad HV (T2 2 Tsurr 2 )(T2 Tsurr )
1(5.67 u 10 8 W/m 2 ˜ K 4 )[(273 5 K ) 2 (273 25 K ) 2 ](273 25 K)(273 5 K )]
5.424 W/m 2 .K
The individual thermal resistances are
Ri
Tf1
T1
R1
Rrad
Tf2
Ro
Rconv,i
R1
1
hi A
1
0.000062 qC/W
(80 W/m 2 ˜ qC)(201.06 m 2 )
r2 r1
(4.015 4.0) m
R sphere
4Skr1 r2 4S (15 W/m ˜ qC)(4.015 m)(4.0 m)
1
ho A
Rconv,o
1
R rad
hrad A
1
(10 W/m ˜ qC)(202.57 m 2 )
1
2
0.000005 qC/W
0.000494 qC/W
0.000910 qC/W
(5.424 W/m 2 ˜ qC)(202.57 m 2 )
1
1
1

o Reqv 0.000320 qC/W
R rad 0.000494 0.000910
1
Reqv
1
Rconv,o
Rtotal
Rconv,i R1 Reqv
0.000062 0.000005 0.000320
0.000387 qC/W
Then the steady rate of heat transfer to the iced water becomes
Tf1 Tf 2
(25 0)qC
Q
64,600 W
Rtotal
0.000387 qC/W
(b) The total amount of heat transfer during a 24-hour period and the amount of ice that will melt during
this period are
Q Q 't (64.600 kJ/s)(24 u 3600 s) 5.581u 10 6 kJ
mice
Q
hif
5.581u 10 6 kJ
333.7 kJ/kg
16,730 kg
Check: The outer surface temperature of the tank is
Q hconv rad Ao (Tf1 Ts )
Q
64,600 W
25qC o Ts Tf1 4.3qC
hconv rad Ao
(10 + 5.424 W/m 2 ˜ qC)(202.57 m 2 )
which is very close to the assumed temperature of 5qC for the outer surface temperature used in the
evaluation of the radiation heat transfer coefficient. Therefore, there is no need to repeat the calculations.
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